Genus 2 curve 834.a.1668.1

• Label: 834.a.1668.1
• Conductor: $834$
• Discriminant: $1668$
• Mordell-Weil group: \(\Z/{8}\Z\)
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + (x^3 + 1)y = -x^2 + x - 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(834\)	\(=\)	\( 2 \cdot 3 \cdot 139 \)
Discriminant:	\( \Delta \)	\(=\)	\(1668\)	\(=\)	\( 2^{2} \cdot 3 \cdot 139 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(372\)	\(=\)	\( 2^{2} \cdot 3 \cdot 31 \)
\( I_4 \)	\(=\)	\(3345\)	\(=\)	\( 3 \cdot 5 \cdot 223 \)
\( I_6 \)	\(=\)	\(401289\)	\(=\)	\( 3 \cdot 7 \cdot 97 \cdot 197 \)
\( I_{10} \)	\(=\)	\(213504\)	\(=\)	\( 2^{9} \cdot 3 \cdot 139 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2$

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{8}\Z\)

Generator	$D_0$						Height	Order
\((1 : -1 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0\)	\(8\)

2-torsion field: 6.0.521667.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 11.76351 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.367609 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 3 T^{2} )\)
\(139\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 12 T + 139 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.60.1	yes

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{USp}(4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).